Improving shortest paths in the Delaunay triangulation

We study a problem about shortest paths in Delaunay triangulations. Given two nodes s, t in the Delaunay triangulation of a point set S, we look for a new point p ∉ S that can be added, such that the shortest path from s to t, in the Delaunay triangulation of S∪{p}, improves as much as possible. We study several properties of the problem, and give efficient algorithms to find such a point when the graph-distance used is Euclidean and for the link-distance. Several other variations of the problem are also discussed.